Hydrodynamics: Viscosity and Diffusion

Hydrodynamics: Viscosity and Diffusion

• Hydrodynamics is the study of mechanics in a liquid, where the frictional drag of the liquid cannot be ignored

• First let’s just consider fluid flow, where the fluid (e.g., water) is treated as continuous

•Can distinguish two types of flow: Steady (time independent) and unsteady (time-dependent, also called turbulent)•A special type of steady flow is laminar – or layered flow

Reynolds Number• R = uL/where u and L are the velocity and

length of the object, and and are the density and viscosity of the fluid

• All macromolecules/bacteria/viruses are in the low R regime where viscous forces dominate

• When modeling the flow of a fluid (water) around such a microscopic object, it is important to consider the boundary layer of fluid near the object – or, its hydration layer

• In physics, the two limits are “stick” and “slip” boundary conditions – with stick conditions appropriate for macromolecules

Hydrodynamic Flow experiments• A number of experimental techniques

involve forcing a macromolecule through a fluid (external force can be electric, gravity, hydrodynamic, or even magnetic)

• In this case we have:

where f is the friction coefficient.• After an extremely short time (~ ps), these

two forces balance and the acceleration goes to zero so that

net externalF F fu ma

0 externalnet

FF or u

f

Friction coefficient• Stokes derived the friction coefficient for a

sphere (w/ stick BC):

where R is the sphere radius• For a few other shaped objects there are

closed expressions for f, but f for a sphere is the minimum value for an equal volume (since f depends mostly on surface area contact with the fluid and a sphere has the minimum surface area for objects of the same volume)

• Rods - f depends on L and axial ratio – Broersma story

• There are now computer modeling programs that treat any shaped object as a collection of spheres and can calculate f

6f R

IgG

lysozyme

Concentration effects on f• Stokes law is valid only in the limit of low

concentration where individual spheres do not “see” each other

• At higher concentrations, flow “wakes” interact with other spheres and increase the friction coefficient, so that to a first approximation:

(1 )of f kc

Viscosity of pure fluid• Definition for laminar flow:

shear stress = F/A (tangential force/area)velocity gradient = du/dy = rate of strain

Shear stress = (rate of strain) – or

F/A = ηo(du/dy) - defines a Newtonian fluid

η units are dyne-s/cm2 = 1 Poise or 1 N-s/m2 = 10 Poise

water = 1 cP at 20oC

Viscous Flow in a cylinder• Laminar flow – velocity profile:

• Flow rate = Volume/time = Q = (PR4/8L) (Poiseuille’s law, 1835; French physician, blood flow)

• Measuring :Q ~ P/ (R,L fixed) and P=gL sotime ~ Then with a standard (water?)

2 2v( ) ( )4

Pr R rL

sample samplesample known

known known

tt

Ostwald viscometer

Viscosity of Solutions of Macromolecules

• Macromolecules distort flow, leading to increased viscosity. Einstein (1906) first solved this problem for spheres:

with = 2.5 for spheres and where is the volume fraction occupied by the macromolecules.

• For other shapes the coefficient, , is larger than 2.5• Other expressions:

relative viscosity = rel = o

or specific viscosity = spec = (o)/o = rel – 1 =

(1 )solution o

Intrinsic Viscosity• Now, = volume of macromolecule/cm3,

but this is equal to , where the partial specific volume is (volume/gm macro) and the concentration is (gm/cm3)

• So, we have , which when extrapolated to c = 0 defines the intrinsic viscosity

vc

vcspecific

0

lim vspecific

c c

Protein M [](cm3/g)Globular Ribonuclease 13,683 3.4

Serum albumin

67,500 3.7

Bushy stunt virus

10,700,000 3.4

Rod-like Myosin 440,000 217

Tobacco Mosaic virus

39,000,000 36.7

Coils PBLG 340,000 184

Measuring Macromolecular Viscosity

• Need low stress, low concentration – • Older method: Ubbelohde glass viscometer• Newer methods:

– Couette viscometer

– Stress rheometer

Example of use of Viscosity Data• First evidence for circular DNA (in T2)

time

A

B

C

Add pancreatic DNAase- induces ss breaks

A single nicks B ds breaks

decreasesC first cut leads to increase, then decrease

F=ma in Diffusion• F(t) = random fluctuating force from solvent collisions (~1016/s at

room T for a 1 m sphere)• We don’t care about details, but want <time averages>

<xF(t)> - f<xu(t)> = m <x a> but <xF> = 0 sonow, let y = x2 and note that

So we get Equipartition of energy says (from thermo, with kB = 1.38x10-23J/K):

<KE> = ½ kBT or

then

2

2

dx d xf x m xdt dt

2 22 2 2( ) / 2 ( )y xx and y xx x or xx y x

2/ 2 / 2f y m y m x

2 21 12 2

BB

k Tm x k T or xm

2 2B Bm y f y k T f y k T

Particle Diffusion• Solution to this is: <y> = (2kBT/f)t = <x2>

A result due to A. Einstein (1905)• So, <x> = 0, but <x2> = 2Dt, where D = kBT/f• In 3-D, since r2 = x2 + y2 + z2 and

<x2>=<y2>=<z2>, we have <r2>=6DtTwenty seconds of a measured random walk trajectory for a micrometer-sized ellipsoid undergoing Brownian motion in water. The ellipsoid orientation, labeled with rainbow colors, illustrates the coupling of orientation and displacement and shows clearly that the ellipsoid diffuses faster along its long axis compared to its short axis.

Second Approach to Diffusion• Instead of looking at a single particle, we can consider

the concentration c(x, y, z)• If we start with a non-uniform initial concentration

profile, diffusion tends to randomize leading to a uniform c

• In 1-D first, introduce the particle flux = J = #/area/timeCan show J = cu, where c = #/volume

[# = cAL, but u=L/t, so J=cAL/(At)=cu]

• Fick’s First Law says J=-D[dc/dx] ; flow ~ c variation(also holds for heat-T, fluid-P, electric current-potential)

AL

Diffusion Equation• But J varies with x and t:

or

• Combining this with Fick’s First law, we get the diffusion eqution:

x x+dx

J(x) J(x+dx)

( ) ( )N J x A t J x dx A tcAdx Adx

( , ) ( ) ( ) ( , )c x t J x J x dx dc J x tt dx dt x

2

2

c J cDt x x

Two Solutions to the Diffusion Eqn.• Solutions depend on initial conditionsA. Narrow band of c at time zero

B. See Figure D3.7 for step gradient initial condition

x=0x

cTime 0 – very sharply peaked

x

c

x=0

Two complications due to Particle Interactions

1. Excluded volume: particles occupy some volume

2. Concentration dependence of f:

Combining these results in:

Note: If c is expressed as a volume fraction, (with ) then for spheres A = 8 and A’ = 6.5

2(1 ...), ,Bk TD Ac Bc where A B are so called virialsf

2(1 ' ' ...)of f A c B c

[1 ...] [1 ( ') ...][1 ' ...]o o

kT Ac kTD A A cf A c f

vc

Why not always work at very low c?1. Some systems are only interesting, or

interact, at higher c2. Need a probe to measure c(x,t): light,

radioactive tracer, fluorescence, etc., and need some threshold signal to detect

3. Some molecules fall apart at very low c – or even denature – e.g. myosin, hemoglobin

Temperature and Solvent Effects• Remember

with both T (K) and η varying with temperature; η varies about 2% per oC for water near 20oC

• With a solvent that includes salts (changing viscosity) we have

• Also, remember that for equivalent sphere f=6R, with R = hydrodynamic radius, including hydration

kT TDf

20, ,,20

293.16( ) o

soln, Twater T soln

water C

D DT K

How to Measure D1. Spreading Boundary Method – used in

ultracentrifuge (see Figure D3.7 again)2. FRAP (Fluorescence Recovery After

Photobleaching) – 3. DLS (Dynamic Light Scattering) – more

later4. NMR (Nuclear Magnetic Resonance) – for

small molecules only – laterTypical D values are ~10-7 cm2/s for small proteins to ~10-9 cm2/s for large ones